Proposition 4

— Let $A$ be a commutative algebra of finite degree over $K$; then $[A: K], \leq [A: K]$ with equality if and only if $A$ is etale.

Let $\Omega$ be an algebraic closure of $K$ and $A?$ the set of algebra homomorphisms of $A$ into $\mathbf{R}$. We have Card $A? = [A: K]$, and $A$ is etale if and only if $A$ is diagonalized by the extension $\Omega$ of $K$ (V, p. 30, Prop. 2). Thus Prop. 4 follows from the Cor. of V, p. 29.